| L(s) = 1 | − 0.295·2-s − 1.91·4-s − 5-s + 3.54·7-s + 1.15·8-s + 0.295·10-s + 0.503·11-s + 3.34·13-s − 1.04·14-s + 3.48·16-s + 2.43·17-s + 4.79·19-s + 1.91·20-s − 0.148·22-s + 8.17·23-s + 25-s − 0.987·26-s − 6.77·28-s − 4.94·29-s − 7.90·31-s − 3.34·32-s − 0.720·34-s − 3.54·35-s + 6.51·37-s − 1.41·38-s − 1.15·40-s + 0.786·41-s + ⋯ |
| L(s) = 1 | − 0.208·2-s − 0.956·4-s − 0.447·5-s + 1.33·7-s + 0.408·8-s + 0.0934·10-s + 0.151·11-s + 0.927·13-s − 0.279·14-s + 0.870·16-s + 0.591·17-s + 1.10·19-s + 0.427·20-s − 0.0317·22-s + 1.70·23-s + 0.200·25-s − 0.193·26-s − 1.27·28-s − 0.918·29-s − 1.41·31-s − 0.590·32-s − 0.123·34-s − 0.598·35-s + 1.07·37-s − 0.229·38-s − 0.182·40-s + 0.122·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.707160881\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.707160881\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 89 | \( 1 + T \) |
| good | 2 | \( 1 + 0.295T + 2T^{2} \) |
| 7 | \( 1 - 3.54T + 7T^{2} \) |
| 11 | \( 1 - 0.503T + 11T^{2} \) |
| 13 | \( 1 - 3.34T + 13T^{2} \) |
| 17 | \( 1 - 2.43T + 17T^{2} \) |
| 19 | \( 1 - 4.79T + 19T^{2} \) |
| 23 | \( 1 - 8.17T + 23T^{2} \) |
| 29 | \( 1 + 4.94T + 29T^{2} \) |
| 31 | \( 1 + 7.90T + 31T^{2} \) |
| 37 | \( 1 - 6.51T + 37T^{2} \) |
| 41 | \( 1 - 0.786T + 41T^{2} \) |
| 43 | \( 1 - 7.85T + 43T^{2} \) |
| 47 | \( 1 + 7.69T + 47T^{2} \) |
| 53 | \( 1 + 9.31T + 53T^{2} \) |
| 59 | \( 1 + 1.91T + 59T^{2} \) |
| 61 | \( 1 + 11.9T + 61T^{2} \) |
| 67 | \( 1 - 2.77T + 67T^{2} \) |
| 71 | \( 1 + 2.31T + 71T^{2} \) |
| 73 | \( 1 - 13.7T + 73T^{2} \) |
| 79 | \( 1 - 0.432T + 79T^{2} \) |
| 83 | \( 1 - 15.7T + 83T^{2} \) |
| 97 | \( 1 - 11.0T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.419622191828918156343510809425, −7.71586257246612835605359224457, −7.43005663189165752648875934934, −6.10534619291780077267869821061, −5.21154281634120617203305202456, −4.81696242471004499670207816416, −3.86028337076090978464886917645, −3.20430840341832637956491664429, −1.60409485023675361194119536887, −0.871275117835323473901073013734,
0.871275117835323473901073013734, 1.60409485023675361194119536887, 3.20430840341832637956491664429, 3.86028337076090978464886917645, 4.81696242471004499670207816416, 5.21154281634120617203305202456, 6.10534619291780077267869821061, 7.43005663189165752648875934934, 7.71586257246612835605359224457, 8.419622191828918156343510809425
